Polygons The word ‘polygon’ is
a Greek word.
PolyPoly means many
and gongon means angles.
Examples of Polygons
Polygons
These are not Polygons
Polygons
Terminology
Side: One of the line segments that make up a polygon.
Vertex: Point where two sides meet.
Polygons
Vertex
Side
Polygons
•Interior angle: An angle formed by two adjacent sides inside the polygon.
•Exterior angle: An angle formed by two adjacent sides outside the polygon.
Polygons
Interior angle
Exterior angle
Polygons
Let us recapitulate
Interior angle
Diagonal
Vertex
Side
Exterior angle
Polygons
Types of Polygons
•Equiangular Polygon: a polygon in which all of the angles are equal
•Equilateral Polygon: a polygon in which all of the sides are the same length
Polygons
•Regular Polygon: a polygon where all the angles are equal and all of the sides are the same length. They are both equilateral and equiangular
Polygons
Examples of Regular Polygons
Polygons
A convex polygon: A polygon whose each of the interior angle measures less than 180°.
If one or more than one angle in a polygon measures more than 180° then it is known as concave polygon. (Think: concave has a "cave" in it)
Polygons
INTERIOR ANGLES OF A POLYGON
Polygons
Let us find the connection between the number of sides, number of diagonals and the number of triangles of a polygon.
Polygons
QuadrilateralPentagon
180o 180
o
180o
180o
180o
2 x 180o = 360o 3
4 sides5 sides
3 x 180o = 540o
Hexagon6 sides
180o
180o
180o
180o
4 x 180o = 720o
4 Heptagon/Septagon7 sides
180o
180o180o
180o
180o
5 x 180o = 900o 5
2
1 diagonal2 diagonals
3 diagonals 4 diagonals Polygons
RegularPolygon
No. of sides
No. of diagonal
s
No. of Sum of the
interior angles
Each interior angle
Triangle 3 0 1 1800
1800/3
= 600
Polygons
RegularPolygon
No. of sides
No. of diagonal
s
No. of Sum of the
interior angles
Each interior angle
Triangle 3 0 1 1800
1800/3
= 600
Quadrilateral
4 1 2 2 x1800
= 3600
3600/4
= 900
Polygons
RegularPolygon
No. of sides
No. of diagonal
s
No. of Sum of the
interior angles
Each interior angle
Triangle 3 0 1 1800
1800/3
= 600
Quadrilateral
4 1 2 2 x1800
= 3600
3600/4
= 900
Pentagon 5 2 3 3 x1800
= 5400
5400/5
= 1080
Polygons
RegularPolygon
No. of sides
No. of diagonal
s
No. of Sum of the
interior angles
Each interior angle
Triangle 3 0 1 1800
1800/3
= 600
Quadrilateral
4 1 2 2 x1800
= 3600
3600/4
= 900
Pentagon 5 2 3 3 x1800
= 5400
5400/5
= 1080
Hexagon 6 3 4 4 x1800
= 7200
7200/6
= 1200
Polygons
RegularPolygon
No. of sides
No. of diagonal
s
No. of Sum of the
interior angles
Each interior angle
Triangle 3 0 1 1800
1800/3
= 600
Quadrilateral
4 1 2 2 x1800
= 3600
3600/4
= 900
Pentagon 5 2 3 3 x1800
= 5400
5400/5
= 1080
Hexagon 6 3 4 4 x1800
= 7200
7200/6
= 1200
Heptagon 7 4 5 5 x1800
= 9000
9000/7
= 128.30
Polygons
RegularPolygon
No. of sides
No. of diagonal
s
No. of Sum of the
interior angles
Each interior angle
Triangle 3 0 1 1800
1800/3
= 600
Quadrilateral
4 1 2 2 x1800
= 3600
3600/4
= 900
Pentagon 5 2 3 3 x1800
= 5400
5400/5
= 1080
Hexagon 6 3 4 4 x1800
= 7200
7200/6
= 1200
Heptagon 7 4 5 5 x1800
= 9000
9000/7
= 128.30
“n” sided polygon
n Association with no. of
sides
Association with no. of sides
Association with no. of triangles
Association with sum of interior
anglesPolygons
RegularPolygon
No. of sides
No. of diagonal
s
No. of Sum of the
interior angles
Each interior angle
Triangle 3 0 1 1800
1800/3
= 600
Quadrilateral
4 1 2 2 x1800
= 3600
3600/4
= 900
Pentagon 5 2 3 3 x1800
= 5400
5400/5
= 1080
Hexagon 6 3 4 4 x1800
= 7200
7200/6
= 1200
Heptagon 7 4 5 5 x1800
= 9000
9000/7
= 128.30
“n” sided polygon
n n - 3 n - 2 (n - 2) x180
0(n - 2) x180
0 / n
Polygons
Septagon/Heptagon
Decagon Hendecagon
7 sides
10 sides 11 sides9 sides
Nonagon
Sum of Int. Angles 900o
Interior Angle 128.6o
Sum 1260o
I.A. 140oSum 1440o I.A. 144o
Sum 1620o I.A. 147.3o
Calculate the Sum of Interior Angles and each interior angle of each of these regular polygons.
1
2 43
Polygons
2 x 180o = 360o
360 – 245 = 115o
3 x 180o = 540o540 – 395 = 145o
y117o
121o
100o
125o
140o z
133o 137o
138o
138o
125o
105o
Find the unknown angles below.
Diagrams not drawn accurately.
75o
100o
70o
w
x
115o
110o
75o 95o
4 x 180o = 720o720 – 603 = 117o
5 x 180o = 900o900 – 776 = 124oPolygons
EXTERIOR ANGLES OF A
POLYGON
Polygons
An exterior angle of a regular polygon is formed by extending one side of the polygon.
Angle CDY is an exterior angle to angle CDE
Exterior Angle + Interior Angle of a regular polygon =1800
DEY
B
C
A
F
12
Polygons
1200
1200
1200
600 600
600
Polygons
1200
1200
1200
Polygons
1200
1200
1200
Polygons
3600
Polygons
600
600
600
600
600
600
Polygons
600
600
600
600
600
600
Polygons
1
2
3
4
5
6
600
600
600
600
600600
Polygons
1
2
3
4
5
6
600
600
600
600
600 600
Polygons
1
2
34
5
6
3600
Polygons
900
900
900
900
Polygons
900
900
900
900
Polygons
900
900
900
900
Polygons
1
23
4
3600
Polygons
No matter what type of polygon we have, the sum of the exterior angles is ALWAYS equal to 360º. Sum of exterior angles
= 360º
Polygons
In a regular polygon with ‘n’ sides
Sum of interior angles = (n -2) x 1800
i.e. 2(n – 2) x right angles Exterior Angle + Interior Angle =1800
Each exterior angle = 3600/n
No. of sides = 3600/exterior angle
Polygons
Let us explore few more problems• Find the measure of each interior angle of a
polygon with 9 sides.• Ans : 1400
• Find the measure of each exterior angle of a regular decagon.
• Ans : 360
• How many sides are there in a regular polygon if each interior angle measures 1650?
• Ans : 24 sides• Is it possible to have a regular polygon with an
exterior angle equal to 400 ?• Ans : Yes
Polygons
Polygons DG